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nLab > nLab General Discussions: Comments on arithmetic geometry

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    • CommentRowNumber1.
    • CommentAuthorandreas
    • CommentTimeJun 6th 2014
    Urs posted on geometric quantization, intermediate Jacobians, and arithmetic geometry, and I just want to add some comments since Arakelov motivic cohomology was mentioned. I wish I could contribute meaningfully to the discussion there, but I don't really understand the question. Those of us who come from arithmetic geometry tend to have trouble absorbing the infinity-categorical point of view, and words like quantization and TQFT's unfortunately don't mean a lot to me (yet!). But coming from the arithmetic side, I share the general feeling that there are great stories here waiting to be uncovered, for someone brave enough.

    As for Arakelov motivic cohomology, my general feeling is that what we did is not as interesting as the more recent stuff of Bunke and Tamme, or the research program of Baptiste Morin on Weil-etale cohomology, which paints a bigger picture than the more narrow story of Arakelov motivic cohomology. Would strongly recommend the articles of Morin on his webpage to anyone interested.

    From my point of view, whenever you want to incorporate arithmetic information into anything else, you run into the problem that we seem to lack the basic tools for making sense of arithmetic schemes and associated cohomological or homotopical invariants. Here are some vague and open questions/remarks, and a lot more could be said!

    1. The category of arithmetic schemes (say regular schemes of finite type over Spec(Z)) is difficult to work with, and it might not be the right category to consider at all! Could there be some other category which is nicer? There is lots of speculation here, mostly motivated by the study of zeta functions (motives, Galois representations, number-theoretic Langlands, Riemann hypothesis, special value conjectures) but not much else. For example, one should probably incorporate some kind of "arithmetic stacks" into the picture.

    2. A key problem, whatever you try to do, is that for the arithmetic theory to make sense, we'd need a good understanding of finiteness properties of our objects. All the time you run into situations where you want to use some cohomology theory (motivic or otherwise) where the coefficients are G_m or some object built out of G_m, like motivic complexes. However, G_m is not constructible/compact in, say, the category of etale sheaves on a fixed arithmetic scheme X, which means that we don't know whether the cohomology groups are finitely generated, and without finiteness statements of this type, much of higher-dimensional arithmetic geometry breaks down completely. Question: Have you encountered questions related to finiteness conditions somewhere in your part of the mathematical universe? I asked a question related to this a long time ago here:
    http://mathoverflow.net/questions/689/finiteness-conditions-on-simplicial-sheaves-presheaves

    3. If you read about the deep ideas of Christophe Deninger and Baptiste Morin, you see formulations of a dream in which there should be a cohomological picture explaining two of the great themes in the study of zeta functions, namely special values (like the Birch-Swinnerton-Dyer conjecture) and zeroes (as in the Riemann hypothesis). One could also ask about a cohomological explanation for a third theme, namely functional equations. In all cases, we don't really have any idea of what's going on, except maybe that the Weil-etale cohomology of Morin et.al. should eventually develop into the right thing for the first issue. However, it is very natural to think that underlying this picture one should have some higher-categorical invariants or structures. You see some traces of this in the work of Morin and Flach, in which they work with classical topoi only. You also see something of this in our Arakelov paper where we work with motivic homotopy theory to make some constructions. However, motivic homotopy theory cannot be the right setting in general because in arithmetic we don't want to invert the affine line.
    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJun 7th 2014
    • (edited Jun 7th 2014)

    Thanks, Andreas,

    I am longing to understand all this at some deeper level. Maybe we may jointly make some progress here.

    Last time that Zoran and David C. kept saying to me (here) something like “you sound like what you need is Arakelov geometry!” I went an tried to look at some introductions. I am afraid I ran out of steam before I was able to get a clear picture though.I should try again.

    One thing that struck me, though, is that what I read Arakelov did very much sounded like… geometric quantization. One picks some 2-form, lifts it to a line bundle etc. Right? (I have to remind myself again. We should write out a definition-section on the nLab!)

    I’ll go and look at the articles by Morin that you are recommending…

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJun 7th 2014
    • (edited Jun 7th 2014)

    A question:

    I have read claims (in various places, but I keep forgetting where, maybe also on your site somewhere?) that modern (Berkovich-style(?)) rigid analytic geometry is to be regarded as superior to traditional algebraic geometry, in that on the one hand it subsumes it, while on the other it generalizes it to situations where methods of algebraic geometry fail. Apparently much of the proofs in local Langlands don’t actually work in algebraic geometry, but crucially go into the analytic regime. (!?)

    Could you remind me of where I might have read this? :-) And more generally, could you (or anyone reading ehre) give me some further pointers to material that would expand on this a bit?

    [edit: stupid me, of course I read that on the nLab! at global analytic geometry – Motivation]

    • CommentRowNumber4.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 7th 2014

    Did you ever see this grant proposal – Homotopy theory and algebraic geometry?

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJun 7th 2014

    Yes, Duisburg-Essen seems to (have) become quite an accumulation point. (I have studied in Essen, actually (before it was fused to Essen-Duisburg)).

    • CommentRowNumber6.
    • CommentAuthorandreas
    • CommentTimeJun 8th 2014
    Thanks David, I hadn't seen that grant proposal! Very interesting.
    • CommentRowNumber7.
    • CommentAuthorandreas
    • CommentTimeJun 9th 2014
    I don't think it's right to say that rigid analytic geometry is superior to algebraic geometry in general. What one could do maybe, is comparing rigid analytic geometry to algebraic geometry in the specific setting where the base scheme is a nonarchimedean field, and in this comparison, it seems likely that the rigid-analytic framework comes out on top for most (all?) purposes. But I'm not an expert!
    • CommentRowNumber8.
    • CommentAuthorandreas
    • CommentTimeJun 9th 2014
    Here are some other references I'm aware of:

    http://notes.andreasholmstrom.org/glossary.php#Adic_spaces

    http://notes.andreasholmstrom.org/glossary.php?l=R#Rigid_analytic_geometry

    http://notes.andreasholmstrom.org/glossary.php?l=R#Rigid_geometry
    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeJun 9th 2014

    re #7: but is the global analytic geometry that Frederic is referring to in that entry not more general than rigid analytic geometry? I thought that was the point, but I don’t really know.

    • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 9th 2014

    There seem to be so many to ways to formulate the non-Archimedean world, and so many ways to compare the ways, such as Ben-Bassat and Kremnizer’s Non-Archimedean analytic geometry as relative algebraic geometry

    In this article, we consider Berkovich analytic spaces from the perspective of algebraic geometry relative to the closed symmetric monoidal categories of Banach spaces. This language is very universal and provides a place to compare different geometries (Huber spaces, rigid analytic spaces, Berkovich spaces and others).

    As they point out, global analytic geometry is supposed to go beyond the non-Archimedean:

    In future work, we intend to embed rigid analytic spaces and Huber spaces into the categories of schemes over the opposite category to Banach algebras. We also intend develop a variant which will work with complete, convex bornological algebras, instead of Banach algebras or Frechet algebras or other types of topological vector spaces. Similarly to Paugam [33] we would like to handle the Archimedean and non-Archimedean cases with a single language.

    Paugam himself writes in Global analytic geometry:

    Many interesting results on polynomial equations can be proved using the mysterious interactions between algebraic, complex analytic and p-adic analytic geometry. The aim of global analytic geometry is to construct a category of spaces which contains these three geometries.

    • CommentRowNumber11.
    • CommentAuthorandreas
    • CommentTimeJun 10th 2014
    Wow, I wasn't aware of these developments. Must read.
    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJun 10th 2014
    • (edited Jun 10th 2014)

    David, thanks for recalling all this.

    Here is the vague picture that I am after, possibly naive:

    Say we start being interested in worldsheets, hence in complex tori. We find the Witten genus by doing differential geometry with a little bit of complex geometry. Then we (or somebody at least…) refine this to the string orientation of tmf which in effect is constructed via fracture squares. By some magic, the original construction in differential geometry transmutes into one in pp-adic geometry.

    While this is understood in a step-by-step fashion, it seems to me that it is generally unclear what the conceptual picture underlying this black magic is. (Or at least so suggests the ratio of votes over answers of this MO question)

    What I would like to see is a unified concept of geometry that makes this black magic become concepually clear. Some unified form of geometry where we start out wanting to compute partition functions of (4k+2)(4k+2)-branes on complex analytic worldvolumes and naturally end up decomposing this problem in a tower of problems in p-adic geometry that also knows about the underlying arithmetic geometry over the corresponding prime fields.

    • CommentRowNumber13.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 10th 2014

    Can you even imagine what’s needed? Would the long sought for pp-adic/adelic cohesion help?

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeJun 10th 2014
    • (edited Jun 10th 2014)

    Rigid analytic spaces locally modeled on Tate algebras/polydiscs will be cohesive, I am pretty sure now. One should check with the experts in suitable words, such as “are rigid spaces locally etale contractible?” (I once sent this question spelled out in some simplicial detail to one of the central experts, but just got a “thanks, but I don’t understand the question” Just using the right words might help.).

    So analytic geometry, archimedean and non-archimedean should be fine. What remains to be understood is how analytification sits in the picture. We want to be talking about arithmetic spaces equipped with differential geometric structure on their analytification. Of course that’s just what Bunke-Tamme do in the archimedean case, but I don’t really understand the organizing principle yet. Something seems to be missing, but likely just on my end.

    When things work out, the following ought to happen: start with a 4k+3 connection on the moduli stack of some differential cohomology theory. Transgress to a complex 2k+1 dimensional space, such that the transgressed connection becomes Kaehler. Now this should automatically know all the positive characteristic, hence arithmetic, geometries underlying this analytic one. Hence the deformation theory will be an Artin-Mazur formal group which in good cases should define a CY-cohomology theory. The corresponding equivariant theory should be the modular functor quantizing the 4k+1 brane.

    For k= 0 we know this is just what happens piecewise, but it would be good to have a more conceptual way to put it all together. As far as the case k=1 and k=2 is understood at all, it seems to naturally want to sit in this picture, too. That’s why I feel this should be the right story, roughly.

    (This here from my phone, bear with me)

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeJun 11th 2014

    Okay, I have tried again to ask now here on MO.

    • CommentRowNumber16.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 11th 2014

    Your ratio of #votes/(# non-self answers) has to be near the highest on MO.

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeJun 11th 2014

    True, recently I wasn’t very lucky with getting replies. On the other hand, I had recently asked several questions of the kind: “this ought to be studied, but has it been studied?” and the number of votes together with the lack of answers is probably the answer itself: “yes, that is indeed an interesting question to ask and, no, it hasn’t been answered yet”.

    Compiling the latest question made me remind myself about Berkovich’s local contractibility statement and I remember now that actually Berkovich spaces are not covered by polydiscs, but by inductive systems of “analytic domains”.

    • CommentRowNumber18.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 4th 2017

    Did you see, Urs, Joe Berner’s edit to his answer to your MO question in #15, announcing a new paper

    • Joe Berner, Etale homotopy theory of non-archimedean analytic spaces, (arXiv:1708.03657)
    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeSep 4th 2017

    Thanks for the pointer! Interesting.

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